Villain is an any-4-cards fish
He likes bluffing when checked to, both as the PFR or caller.

My issue is regarding the turn, though.
I like leading turn with my hand since it's possible to push him off some weak hands
that have good equity against me and that he could decide to check back.

River, is this a default shove, once he only calls the turn?
The only reason for a shove would be to make him fold a set though.
He will not raise a set on the turn, for sure.
There is a chance he also calls with worse two pair, although I'm not very optimistic about it.